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The paper provides several original conditions involving ranks and traces of functions of a pair of orthogonal projectors (i.e., Hermitian idempotent matrices) under which the functions themselves are orthogonal projectors. The results are established by means of a joint decomposition of the two projectors. [ABSTRACT FROM AUTHOR]

Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore- Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares. [ABSTRACT FROM AUTHOR]

The article describes the numerical range of a 3 x 3 magic square. It mentions that numerical range, the field of values of an n x n matrix A, of a matrix is a subset of the complex plane and it provides information about the eigenvalues and the eigenspace of the matrix. The results were applied to the square, the Luoshu and found that its numerical range is a piece of cake almost. Several numerical calculations and diagrams are also presented to elaborate this.

In this article a new parameterization of magic squares of order three is presented. This parameterization permits an easy computation of their inverses, eigenvalues, eigenvectors and adjoints. Some attention is paid to the Luoshu, one of the oldest magic squares. [ABSTRACT FROM AUTHOR]

Abstract: A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF † and F † F, determined by a complex square matrix F and its Moore–Penrose inverse F †. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rakočević [J. Benítez, V. Rakočević, Matrices A such that AA † − A † A are nonsingular, Appl. Math. Comput. 217 (2010) 3493–3503]. Further characteristics of FF † and F † F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF † and F † F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein. [Copyright &y& Elsevier]

A general class of matrices, covering, for instance, an important set of proper rotations, is considered. Several characteristics of the class are established, which deal with such notions and properties as determinant, eigenspaces, eigenvalues, idempotency, Moore-Penrose inverse, or orthogonality. [ABSTRACT FROM AUTHOR]

Abstract: By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility. [Copyright &y& Elsevier]